A Well-balanced and Positivity-preserving Godunov-type Numerical Scheme for a Spray Model

Computational exact Riemann solvers are constructed for a spray model. The model is characterized by a system of balance laws with nonconservative terms in the equations for conservations of momentum and energy. The proposed scheme is combined from an upwind scheme for the equation of the volume fraction evolution and a Godunov-type scheme for the governing equations. The numerical results show that the scheme provides us with the convergence and it possesses a good accuracy, even when resonant phenomena occur. It is interesting to see that the proposed scheme is well-balanced, and preserves the positivity of density, of volume fraction, and the C-property.